Question

Can anyone come up with a way to prove this? i have been working on it since yesterday and can not figure out how to start :/ Prove that if V ⊂ R^n is spanned by a set of vectors S, then a vector w⃗ ∈ R^n is orthogonal to every vector in V if and only if w⃗ is orthogonal to every vector in S.

Answer 1

Suppose not. Then there exists wbox such that it is orthogonal to every vector in V but not orthogonal to vector S1 in S. However, since S spans V, a vector kS1 (some constant times S1) is in V and is not orthogonal to wbox. Contradiction. The proof of the converse is similar.